A Note on Hardness of Computing Recursive Teaching Dimension

TL;DR

This paper demonstrates that computing the recursive teaching dimension (RTD) for a concept class is computationally hard, requiring super-polynomial time under the exponential time hypothesis, matching the brute-force algorithm's complexity.

Contribution

It establishes a tight lower bound on the computational complexity of calculating RTD, showing it is unlikely to be efficiently solvable.

Findings

Computing RTD requires $n^{ ext{Omega}( ext{log } n)}$ time under ETH.

The lower bound matches the brute-force algorithm's running time.

RTD computation is computationally hard under standard complexity assumptions.

Abstract

In this short note, we show that the problem of computing the recursive teaching dimension (RTD) for a concept class (given explicitly as input) requires $n^{\Omega(\log n)}$-time, assuming the exponential time hypothesis (ETH). This matches the running time $n^{O(\log n)}$ of the brute-force algorithm for the problem.